Not sure how to solve this at all...
If $\widehat{a}, \widehat{b}, \widehat{c}$ are 3 unit vectors such that the measure of the angle between $\widehat{a}, \widehat{b}$ equals that between $(\overrightarrow{a}\times \overrightarrow{b}) , \overrightarrow{c} = \theta$ . Prove that $$(\overrightarrow{a}\times \overrightarrow{b}) \cdot \overrightarrow{c} = \frac{1}{2}\sin (2\theta )$$
You have $$(a\times b)\cdot c=|a\times b||c|\cos \theta=|a||b|\sin\theta|c|\cos\theta=\sin\theta\cos\theta=\frac 12\sin2\theta$$